IEEE TRANSACTIONS ON NEURAL NETWORKS, …web.cse.ohio-state.edu/~wang.77/papers/CWJ.tnn04.pdfIEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 5, SEPTEMBER 2004 1027 Synchronization

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 5, SEPTEMBER 2004 1027

Synchronization Rates in Classes ofRelaxation Oscillators

Shannon R. Campbell, DeLiang Wang, Fellow, IEEE, and Ciriyam Jayaprakash

Abstract—Relaxation oscillators arise frequently in physics, elec-tronics, mathematics, and biology. Their mathematical definitionspossess a high degree of flexibility in the sense that through appro-priate parameter choices relaxation oscillators can be made to ex-hibit qualitatively different kinds of oscillations. We study numer-ically four different classes of relaxation oscillators through theirsynchronization rates in one-dimensional chains with a Heavisidestep function interaction and obtain the following results. Relax-ation oscillators in the sinusoidal and relaxation regime both ex-hibit an average time to synchrony, , where is thechain length. Relaxation oscillators in the singular limit exhibit

, where is a numerically obtained value less than 0.5.Relaxation oscillators in the singular limit with parameters modi-fied so that they resemble spike oscillations exhibit log( )in chains and log( ) in two-dimensional square net-works of length . Finally, using a sigmoid interaction results in

2, for relaxation oscillators in the sinusoidal and relax-ation regimes, indicating that the form of the coupling is a control-ling factor in the synchronization rate.

Index Terms—Coupled oscillators, neural dynamics, relaxationoscillators, synchronization rate, synchrony.

I. INTRODUCTION

THE STUDY of coupled oscillators reveals a rich diversityof behaviors and has many ties to physical, chemical, and

biological phenomena [24], [49]. The study of how coupled os-cillators achieve synchrony is important because of its relevanceto experimental observations of synchronous neural firing pat-terns of various mammalian, insect, and reptilian species [27],[33], [35], [38].

The phrase “relaxation oscillations” was coined by Van derPol in 1926 in his analysis of a triode circuit [44]. A relaxationoscillation is described by two distinct time scales: a slow timescale, which originally reflected the charging of a capacitor, anda fast time scale, which described a quick discharge. In the equa-tions that describe a relaxation oscillator, the two time scalesare typically based on a single parameter . As , the timescales become more disparate.

Relaxation oscillations were immediately recognized ashaving similarities to biological oscillations. Van der Pol and

Manuscript received June 13, 2003; revised January 16, 2004. This work wassupported in part by National Science Foundation Grant IIS-0081058 and in partby the Air Force Office of Scientific Research Grant F49620-01-1-0027.

S. R. Campbell is with the Clinical Center, Diagnostic Radiology Depart-ment, National Institutes of Health, Bethesda, MD 20892 USA (e-mail: [email protected]).

D. Wang is with the Department of Computer and Information Science andCenter for Cognitive Science, The Ohio State University, Columbus, OH 43210USA.

C. Jayaprakash is with the Department of Physics, The Ohio State University,Columbus, OH 43210 USA.

Digital Object Identifier 10.1109/TNN.2004.833134

van der Mark may have been the first to use relaxation oscil-lations to model biological phenomena, the heart beat [45].In 1952, Hodgkin and Huxley gave a mathematical model ofthe membrane potential and ionic conductances of a nervecell using a four variable system of differential equations [19]and these were later simplified to a two variable system ofequations that is essentially a relaxation oscillator [16], [32].Later, Mayeri [28] derived the Van der Pol relaxation oscillatoras a quantitative description of his experiments with the car-diac ganglion cells of the lobster. Morris and Lecar [30] alsoderived a two variable relaxation oscillator in their study of theconductances and currents in the barnacle giant muscle fiber.Due to these direct links to physiology, relaxation oscillatorshave been frequently used as models of biological behavior[17], [34], [39], [43], [47].

Synchrony in networks of locally coupled relaxation oscilla-tors has been studied previously. Somers and Kopell [39] notedthat synchrony occurs more rapidly in relaxation oscillatorchains as decreases. They explained this quick synchro-nization for small using “fast threshold modulation” andconjectured that the average time to synchrony increaseslinearly with , the number of oscillators in a chain. However,it is not known whether this scaling relation is because theoscillations are of relaxation type, or because of the particularinteraction used. We present numerical evidence indicatingthat the linear scaling relation persists when using a Heavisidecoupling even when the relaxation oscillators are clearly inthe sinusoidal regime, and that relaxation oscillators in thesinusoidal and relaxation regime coupled with a smoothlyvarying sigmoid function exhibit , indicating that theform of the interaction is a critical factor in the synchronizationrate. This complements Daido’s work [10], indicating that adiscontinuous interaction leads to partial perfect synchrony inglobally coupled phase oscillator networks with heterogeneousfrequencies, and is consistent with simulations and statementsmade by Somers and Kopell [40] indicating that increasing thesteepness of the sigmoid coupling function aids synchroniza-tion.

Terman and Wang [43] proved that networks of locally cou-pled identical relaxation oscillators with a Heaviside type inter-action, and as , can achieve synchrony at an exponentialrate independent of the number of oscillators or the dimensionof the network. However, this rate of synchronization is possibleonly when the oscillators are initially on a specific portion of thelimit cycle. We study the case in which the initial conditions ofthe oscillators are randomly and uniformly distributed in a boxthat bounds the limit cycle and observe different results, i.e.,

and the observed values of are less than 0.5.

1045-9227/04$20.00 © 2004 IEEE

1028 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 5, SEPTEMBER 2004

Campbell et al. [7] gave heuristic arguments and numericalevidence that chains of identical spike oscillators achieve syn-chrony at times proportional to the logarithm of the system size.They also presented simulations indicating thatfor locally coupled two-dimensional (2-D) square networks oflength . We examine relaxation oscillators whose parametershave been adjusted so that they behave qualitatively as spike os-cillators and find similar synchronization rates in both 1-D and2-D networks.

A relaxation oscillator is highly flexible and its parame-ters can be adjusted so that the type of oscillation it exhibitssmoothly varies from sinusoidal to spike oscillations. Webriefly define four different classes of oscillations that relax-ation oscillators can manifest. For relaxation oscillatorsexhibit sinusoidal oscillations; they can have an almost circularlimit cycle and a sinusoidal waveform. We refer to this caseas a relaxation oscillator in the sinusoidal regime. As de-creases, the waveform takes on a square-wave appearance. Theoscillator spends most of its period on the two branches of the

nullcline and quickly switches from one branch to another.These are relaxation oscillations and we refer to oscillators inthis parameter regime simply as relaxation oscillators in the re-laxation regime. The third type of oscillation occurs withand the time spent switching between the two branches is zero.We refer to this as a relaxation oscillator “in the singular limit”[5], [25]. The fourth type of oscillation is also in the singularlimit, but with parameters modified so that the oscillator spendsa vanishingly small portion of its period on the right branch.We refer to this as a relaxation oscillator in the spiking regime.

While these four classes of oscillations have been createdbased on the intuition that they have meaningful distinctions,to our knowledge, no objective measure of difference has previ-ously been applied. In this paper, we examine how the averagetime to synchrony varies with chain length and find three dif-ferent synchronization rates.

The form of the coupling is also an important factor in thesynchronization rate. We explore two different types of cou-pling, the Heaviside step function and a smooth sigmoid, andfind that they yield two different synchronization ratesand , respectively, for relaxation oscillators in the si-nusoidal and relaxation regime.

This paper is organized as follows. In Section II, we detailhow to modify the parameters of the defining equations ofa Terman–Wang oscillator [43] to achieve the four differentclasses of relaxation oscillators discussed above. In Section III,we describe some basic properties of pairs of relaxation os-cillators as well as some properties of relaxation oscillatorsin the singular limit. In Section IV, we discuss chains of re-laxation oscillators. In Section V, we present results for therate of synchronization in the spiking regime. In Section VI,we compare spike oscillators with relaxation oscillators inthe spiking regime. In Section VII, we show our data for thesynchronization rate in the singular limit. In Section VIII, wepresent results for the synchronization rates in the relaxationand sinusoidal regime with Heaviside and sigmoid coupling. InSection IX, we give results in 2-D networks in the singular limitand in the spiking regime. Discussions are given in Section X.

Fig. 1. (a) Limit cycle (thick curve) for system (1) with " = 1:0. The x- andy nullclines are the thinner solid and dashed curves, respectively. (b) Temporalevolution of the x variable and y variable. The parameters are � = 3, = 42,and � = 10.

II. DIFFERENT CLASSES OF RELAXATION OSCILLATORS

We examine a Terman–Wang relaxation oscillator [43] whichis simplified from the Morris–Lecar model of neural behavior[30]

(1a)

(1b)

Fig. 1(a) displays the limit cycle for (1), using , and it doesappear qualitatively sinusoidal, as does the temporal evolutionof both variables in Fig. 1(b). As decreases, the system changessmoothly from sinusoidal oscillations to relaxation oscillations.In Fig. 2(a)–(c), we display limit cycles and temporal evolutionof the variables for progressively smaller values of ( ,

, ). As becomes smaller, the limit cycle movescloser to the cubic nullcline, . An oscillator travelsalong one branch of the cubic until it reaches the knee (a localextremum) at which point its velocity is dominated by motionalong the direction and it quickly “jumps” to the other branchof the cubic, where it resumes its relatively slower travel alongthat branch.

We now describe how trajectories for coupled relaxation os-cillators are analytically computed in the singular limit, .In this regime, the motion of an oscillator is determined by asingle variable and knowledge of which branch the oscillator ison (see [43]). The fast system of (1) is obtained by setting ,which results in

(2a)

(2b)

CAMPBELL et al.: SYNCHRONIZATION RATES IN CLASSES OF RELAXATION OSCILLATORS 1029

Fig. 2. Three different limit cycles for system (1) and the temporal plots oftheir x- and y-variables are shown for three different values of ". (a) " = 0:33.(b) " = 0:1. (c) " = 0:01. The x- and y nullclines are the thinner solid anddashed curves, respectively. The parameters are � = 3, = 42, and � = 10.

The slow system for (1) is derived by introducing a slow timescale and then setting . The evolution of an oscil-lator on the left branch in the slow time scale is

(3a)

(3b)

where describes the left branch of (1). Becauseand , we approximate (3b) with

(4)

For the right branch, these same steps result in

(5)

The set of (2)–(5) describe a relaxation oscillator in the singularlimit.

The equations describing along the left and rightbranches in the slow time scale are readily solved analyticallyand many interesting quantities can be computed. For example,we compute the total period of oscillation, , the time neededto traverse both branches. The time it takes to travel along theright branch from the position of the left knee tothe position of the right knee is given by

(6)

The time needed to travel along the left branch, , fromto , is given by

(7)

and .The parameters and control the height of the smoothly

varying portions of the nullcline. We fix the values andand use , , and to control the velocity of an oscillator

on either branch. Through reduction of the time an oscillatorspends on the right branch in comparison to the time it spendson the left branch, we can make the relaxation oscillator mimica spike oscillator.

We define the branch ratio, , as the time an oscillatorspends on the right branch divided by the time it spends on theleft branch

(8)

is an analog to the duty cycle. For small , the oscillationcontains an infinitesimal firing time, or voltage spike. We denoterelaxation oscillators with as in the spiking regime.

III. PAIRS OF RELAXATION OSCILLATORS

We describe briefly how pairs of relaxation oscillators syn-chronize and discuss other known solutions. We use a pair ofcoupled relaxation oscillators given by

(9a)

(9b)

(9c)

(9d)

(9e)

The parameter is the coupling strength and is positive. Theinteraction term is a sigmoid, mimicking excitatory chemicalsynapses. Increasing elevates the nullcline .This is a property seen in descriptions of neural behavior [16],[19], [30], [48]. When the parameter , the interactionapproximates a Heaviside step function. The threshold of theinteraction term is placed between the left and right branchesof the nullcline, thus, the interaction term is either on or offdepending on whether or not an oscillator is on the right or leftbranch.

When the leading oscillator jumps up from the left branch tothe right branch, it is said to “fire.” At this time, the interactionterm becomes nonzero and excitation is sent to the second os-cillator. If the second oscillator is positioned on the left branchslightly above the left knee, then the excitation will raise thenullcline of the second oscillator and induce it to fire. As theoscillators travel along the right branch, the Euclidean distancebetween them decreases. An equivalent dynamic occurs whenthe oscillators jump down. This cycle repeats and the Euclideandistance between the oscillators decreases exponentially to zero.If the leading oscillator does not induce the second oscillator tojump immediately, then other trajectories occur, many of whichalso lead to a synchronous solution [39], [43].


Fig. 3. Plot of the nullclines and the synchronous limit cycle of a pair ofrelaxation oscillators as defined in (9). The dotted cubics are the excited andunexcited x nullclines, and the dash-dot curve is the y nullcline. The thick solidcurve is the synchronous limit cycle for a pair of oscillators and is the resultof numerical calculation. The parameters are � = 2, � = �0:5, � = 5000,� = 8, = 12, " = 0:005, and � = 1000. Upper left branch (ULB), lowerleft branch (LLB), upper right branch (URB), lower right branch (LRB), LKis the y position of the left knee, RK is the y position of the right knee, and� is the coupling strength.

Synchrony can be defined in many ways. In this paper, weuse two different measures of synchrony. In the singular limit,the firing time is precisely defined and a network is consideredsynchronous when the entire chain of oscillators first jumps upor down simultaneously. When , we sum over the Eu-clidean distances squared between all pairs of oscillators anddivide by the number of pairs to create a measure of the averagedistance squared between oscillators, . The oscillators aresynchronous when is less than a threshold value. We testedseveral other measures of synchrony (data not shown) and foundno significant difference in our results.

The interaction term alters the limit cycle of the oscillators.When both oscillators jump up from the lower left knee simul-taneously, the interaction term raises the nullclines of bothoscillators and they must travel an extra distance in the di-rection. This same additional distance must be traversed whenboth oscillators jump to the left branch, which is then loweredby an amount . These altered nullclines are referred to as theupper and lower left (or right) branches. In Fig. 3, we displaythe synchronous limit cycle of system (9).

The alteration of the limit cycle lengthens the period of thesynchronous solution, . The time needed to traverse the upperright branch is

(10)

and the time needed to travel from to , along thelower left branch, is

(11)

The synchronous period is , and the branchratio becomes .

Under certain conditions, a pair of relaxation oscillators canhave stable antiphase solutions [23]. In [6], the minimum cou-pling strength for a pair of coupled Terman–Wang oscillators isderived such that if the coupling strength is below this minimumvalue, then both antiphase and synchronous solutions occur de-pendent on the initial conditions. The antiphase solution has a

different limit cycle and a different period than the synchronoussolution.

We now define the compression ratio , which is a qualita-tive measure of how the time difference between two oscillatorsdecreases during a period. The time difference is analogous toa phase difference and is defined when two oscillators are onthe same branch. Specifically, is the ratio of the initial timedifference to the time difference after the two oscillators havejumped up and down together. This ratio is computed using themaximal distance between the oscillators for an initial jumpfrom the left branch, . While this specific case rarely occurswhen using random initial conditions, it is an estimate of thecompression per cycle. for system (9) is

(12)

where the values of are given by

(13)

It can be shown that (12) has its maximum value (fastest syn-chronization) when , i.e., when the amount of time anoscillator spends on the left branch is equal to the amount oftime it spends on the right branch.

IV. CHAINS OF RELAXATION OSCILLATORS

We define a chain of relaxation oscillators as follows

(14a)

(14b)

(14c)

The coupling strengths are normalized using

(15)

matrix represents nearest neighbor coupling

if

if (16)

and is the number of nearest neighbors that oscillator has.Both and vary between 1 and . except forand , where . This normalization ensures that alloscillators have the same trajectory in phase space when syn-chronous regardless of how many neighbors they have [46].

We briefly describe how to simulate networks of oscillatorsin the singular limit. In Section III, we showed how the periodcould be exactly calculated by determining the time spent trav-eling each branch. Similarly, given an initial position, the timeit takes an oscillator to travel to a local extremum of the cubicand jump can be computed analytically [see (4) and (5)]. For


a system of two or more oscillators, we identify the oscillatorthat will next jump and at what time it will jump. All oscillatorsare updated to this time. The interactions between the oscillatorthat jumped and its neighbors are modified, and this step repeatsuntil no jump occurs. The oscillator to next jump is determinedand the above procedure begins again. This algorithm is calledthe singular limit method, is described in [25], and is used inall our simulations in the singular limit. It allows us to simulatelarge numbers of oscillators efficiently. In the non-singular limit, we use a Runge–Kutta integration method andpresent data from smaller chain lengths .

Similar to pairs of relaxation oscillators, in chains of relax-ation oscillators, the coupling strength needs to be greater than aspecific value, otherwise both desynchronous and synchronoussolutions are possible depending on initial conditions. This min-imum coupling strength for Terman–Wang oscillator chains isderived in [5]. We focus on the properties of synchronization,and for all results in this paper, we use a coupling strength largerthan the minimum coupling strength.

Also, we believe that in one-dimensional (1-D) networks ofidentical relaxation oscillators (14), traveling waves can onlyarise if the topology of the network is a ring. Traveling waves inrings are detailed in [5] and basically require that the oscillatorshave a specific order and specific temporal relationships so thatthey lead one another around the limit cycle. The time differencebetween the oscillators is limited by the amount of time spenton the right branch (because at least two or more oscillatorsmust be on the right branch at all times to maintain the travelingwave). As the time spent on the right branch becomes small,a traveling wave must consist of a large number of oscillatorsevenly positioned on the limit cycle, and thus, the conditionsfor creating a traveling wave become a small volume of phasespace. Also, traveling waves are thought to occur only in ringtopologies, in which the ordering can be preserved. At the chainends, it seems unlikely that the wave is able to reflect, or in otherwords, reverse the ordering of all oscillators in the chain.

Because the coupling strength used in all simulations pre-sented in this paper is above the limit required for one knowntype of antiphase solution (described in [5]), and because wedoubt that traveling waves can exist in chains, we believe thatsynchrony is the only solution available for system (15). How-ever, we can not rule out the possibility of other nonsynchronoussolutions. In our simulations, we did not observe any other finalstate of the system besides synchrony.

V. SYNCHRONIZATION IN CHAINS OF RELAXATION

OSCILLATORS IN THE SPIKING REGIME

Fig. 4 displays as a function of for chainlengths that vary from 2 to , with . Ourresults are consistent with an average time to synchrony thatincreases as the logarithm of the system size. This same syn-chronization rate was observed in simulations of spike oscillatorchains with pulsatile coupling [7]. Only several hundred trialsare needed to compute the averages because the distribution ofthe synchronization times does not have a long tail (an exampledistribution is shown in Fig. 11 and discussed in Section VII).

Fig. 4. Average time to synchrony divided by the period of the synchronoussolution, hT i=� , as a function of log (n) for chains of relaxation oscillatorsin the spiking regime, (B = 8:4 � 10 ). The error bars are shown andindicate the standard deviation. The parameters are � = 2079, = 2082, and� = 3:5. the averages are based on 500 trials, except for the last two pointsrepresenting 50 000 and 10 oscillators, which are based on 100 and 10 trials,respectively. The initial conditions for each trial were uniformly and randomlydistributed in the box that bounds the limit cycle.

However, the data in Fig. 4 could also result from a power-lawrelationship, , with , which may be likely con-sidering that relaxation oscillator chains in the singular limit(Section VII) exhibit this scaling relationship. We cannot de-termine from our numerical results which scaling relationshipmore accurately describes relaxation oscillators in the spikingregime.

VI. COMPARISON OF RELAXATION OSCILLATORS IN THE

SPIKING REGIME WITH SPIKE OSCILLATORS

A. Comparison of Oscillator Pairs

Relaxation oscillators are typically thought to have muchin common with spike oscillators. Both oscillators are used tomodel neuronal behavior and both are frequently examined witha discontinuous interaction. We make qualitative comparisonsof oscillator dynamics for a pair of Terman–Wang relaxationoscillators and a pair of spike oscillators used in [7].

A network of spike oscillators is defined as

(17)

The parameter and the threshold control the period of an un-coupled oscillator. The threshold of an oscillator is 1. When

the oscillator is said to “fire;” its value is instantly resetto 0 and it sends excitation to its neighbors. represents the

firing times of oscillator and is the Dirac Delta func-tion. The coupling term, , is the same as defined in (15).When oscillator fires at time , oscillator receives an in-stantaneous pulse, which increases by . If increasesabove threshold, then it fires and its value is reset as follows,

. Since oscillator fires, immedi-ately oscillator receives excitation and, thus, .This particular realization of a network of spike oscillators wascalled “Model A” in [20].

When one relaxation oscillator fires, it can induce its neighborto immediately fire, a dynamic that also occurs in spike oscilla-tors. However, in relaxation oscillators, the interaction is of fi-nite duration, unlike the instantaneous interaction between spikeoscillators. Further, in relaxation oscillators one oscillator can


Fig. 5. (a) Return map for a pair of relaxation oscillators. The horizontalaxis represents the initial time difference, t , between the oscillators on thelower left branch and the vertical axis represents the time difference betweenthe oscillators when they are next on the lower left branch, t . (b) Plot of thenumber of periods needed before both oscillators are in the jumping region C ,as a function of the initial time difference between the two oscillators t . Theparameters are � = 2079, = 2082, and � = 3:5.

receive excitation for a short time, which alters the branch theoscillator is on, but when excitation ceases, the position of theoscillator can be identical to what it would have been withoutexcitation, unlike spike oscillators with pulsatile coupling. Also,the interaction increases the synchronous period for relaxationoscillators, while decreasing it for spike oscillators.

We calculate a return map [Fig. 5(a)] for a pair of relaxationoscillators initially placed on the lower left branch. The initialtime difference between the two oscillators is represented onthe horizontal axis and the vertical axis displays their time dif-ference when they are next on the lower left branch. This returnmap ignores changes in ordering. We note several features ofthis return map. The “jumping region” corresponds to those ini-tial conditions in which the firing of one oscillator immediatelyinduces the other to fire. For the parameters used, the jumpingregion represents the first 70% of the initial conditions. Just be-yond the jumping region, there is an initial time difference, ,that results in perfect synchrony. Somers and Kopell [39] calledthe region near this time difference “super compressed,” becauselarge initial time differences are mapped into small time differ-ences in just one period. For larger initial time differences, theleading oscillator can jump up and down before the second os-cillator can jump up. For the parameters we use, this trajectoryalso significantly reduces the time difference. In Fig. 5(b), wedisplay the number of periods needed before two oscillators arein the jumping region, , as a function of their initial time dif-ference. The maximum is one.

For a pair of spike oscillators, we display the return map andin Fig. 6(a) and (b), respectively, (from [7]). The parameters

are such that the jumping regions comprise approximately 70%of the limit cycle. The spike oscillator pair exhibits an unstablefixed point and many cycles can pass before the oscillators firesynchronously depending on their initial conditions.

B. Comparison of Oscillators Chains

In Fig. 7, we display as a function of forchain lengths from 5 to for spike oscillators and for relax-ation oscillators modified so that . The average synchro-nization time is not noticeably different than with

(data shown in Fig. 4).We tried to compare oscillators that exhibit the greatest simi-

larity. We chose parameters so that the size of the jumping regionis approximately 70% of the limit cycle for both oscillators. Our

Fig. 6. (a) Return map for two pulse coupled spike oscillators. The phasedifference between the oscillators before (� , horizontal axis) and after (� ,vertical axis) they have jumped. (b) Plot of the number of cycles needed, C ,before the oscillators are synchronous as a function of � . Figure taken from[7].

Fig. 7. Plot of hT i=� as a function of log (n) for relaxation oscillatorchains (plus signs) with an instantaneous traversal of the right branch and spikeoscillator chains (diamonds). Both oscillators have jumping regions which areapproximately 70% of the limit cycle. The data for the spike oscillators is from[7] and the parameters for the relaxation oscillators are � = 3:5, � = 4, and = 7. The averages are based on approximately 300 trials except for the pointrepresenting a chain length of 10 relaxation oscillators, which is based on 25trials. The initial conditions for the relaxation oscillators were uniformly andrandomly distributed on the lower left branch, since it is not possible to placean oscillator on the right branch in this case.

numerical results are consistent with a time to synchrony that in-creases as the logarithm of the chain length for spike oscillatorsand relaxation oscillators with . However, relaxationoscillator chains achieve synchrony faster than spike oscillatorchains. This difference may be qualitatively explained by thereturn map for two relaxation oscillators [Fig. 5(a)] and that fortwo spike oscillators [Fig. 6(a)]. The return map for relaxationoscillators indicates a generally larger compression per periodand the number of cycles needed before both oscillators jumptogether is only one [Fig. 5(b)], while for two spike oscilla-tors can be greater than one [Fig. 6(b)]. Based on these returnmaps, one might expect that the time needed to synchronize achain of relaxation oscillators would be less than that for a chainof spike oscillators.

In networks of spike oscillators, clusters of synchronous os-cillators form and the evolution of the network is such that thenumber of domain walls does not increase [7]. Assuming thatthe rate at which clusters merge is a constant that is depen-dent only on the system parameters, then the number of clustersshould decrease exponentially, and this is what is observed nu-merically [7]. For relaxation oscillators in the spiking regime,we believe that similar dynamics occur. Our preliminary inves-tigations indicate that occasionally, a small number of oscilla-tors switch from one cluster to another, but the formation of newclusters is not observed. If the rate at which clusters merge is a


Fig. 8. Plot of hT i=� as a function of log (n) for relaxation oscillatorchains in the singular limit. The four different symbols represent four differentvalues of B . The parameters are shown in Table I. The initial conditions wererandomly and uniformly distributed within the box bounding the limit cycle.The data for B = 8:4� 10 are described the Fig. 4 caption. For the otherthree values of B , the averages are based on 250 trials each, except for the datapoints representing 10 oscillators, which are based on 10 trials each.

TABLE ITHE BRANCH RATIOS AND THE PARAMETERS FOR THE DATA

DISPLAYED IN FIG. 8

constant that depends only on system parameters then we canexpect exponential growth of the cluster size.

VII. SYNCHRONIZATION IN CHAINS OF RELAXATION

OSCILLATORS IN THE SINGULAR LIMIT

In Fig. 8, we display as a function of for re-laxation oscillators with several different ( ,

, , and ) for chain lengths thatvary from 2 to . The parameters are listed in Table I and werechosen to examine how the average time to synchrony changesas the relaxation oscillators move out of the spiking regime. Thedata show that chains with have the same average syn-chronization time. For larger chain lengths, networks with dif-ferent parameters exhibit different times to synchrony. The di-amonds , plus signs , andsquares all appear to lie on straight lines for valuesof and are, thus, consistent with . Thedata for (denoted by ) are no longer linear on thissemilog plot.

The parameters used in Fig. 8 increased the amount of timean oscillator spent on the right branch while attempting to min-imize changes in other aspects of the oscillator. Specifically,the distance between the right branch and the nullcline for

decreased while the distance between the left branchand the nullcline for remained constant. This changedecreased the speed of the oscillator on the right branch whilenot affecting the speed of the oscillator on the left branch. The

Fig. 9. Plot of hT i=� as a function of log (n) for relaxation oscillatorchains in the singular limit. The five different symbols represent relaxationoscillator chains with five different values of B . The averages are based on250 trials each, except for the data point representing 10 oscillators, which isbased on 10 trials. The parameters are shown in Table II. The initial conditionswere randomly and uniformly distributed within the box bounding the limitcycle.

TABLE IIBRANCH RATIOS AND THE PARAMETERS FOR THE DATA DISPLAYED IN FIGS. 9

AND 10. THE LAST COLUMN CONTAINS THE SLOPES OF THE LINES

CALCULATED FROM THE DATA SHOWN IN FIG. 10

coupling strength was not altered and the position of the syn-chronous limit cycle remained the same. The period increasesby a factor of two as goes from zero to one.

The slopes of the lines for , ,and decrease as the oscillator spends more time onthe right branch, indicating a faster approach to synchrony. Thismay be qualitatively explained by noting that the compressionratio, , increases as increases.

With the data in Fig. 8 indicate that no longerincreases linearly with . This leads us to examine largervalues of . In Fig. 9, we display as a function of

with branch ratios larger than those shown in Fig. 8,specifically, , , , ,and ; the trend that started in Fig. 8 becomes morepronounced. The parameters are listed in Table II.

In Fig. 10, we display the same data shown in Fig. 9 exceptthat the data are plotted on a log-log graph. Also, we do notshow all of the data from Fig. 9 because the average times tosynchrony for do not yield straight lines and we as-sume that they do not reflect the asymptotic behavior of thesystem. For system sizes from – , our data is consistentwith . The slopes of the lines were obtained using aleast squares fit and are listed in Table II. They vary from 0.46

to 0.14 .In Fig. 10, the data for is shown in the inset along

with error bars representing the addition and subtraction of thestandard deviation. The standard deviations for other parametersare not shown, but are similar to those seen in the inset.


Fig. 10. Log-log plot of hT i=� as a function of n for the data shown inFig. 9. The straight line represents a least squares fit to the data for B = 0:53.A plot of hT i=� and its standard deviations for B = 0:14 are shown in theinset. The parameters used and the slopes of the lines computed for each valueB of are shown in Table II.

Fig. 11. Histogram of hT i=� for 25 075 trials of a chain of 500 relaxationoscillators in the singular limit, B = 0:53, with parameters � = 3:5, � = 3,and = 6. The histogram indicates that the distribution of synchronizationtimes does not have a long tail. The initial conditions were randomly anduniformly distributed within the box bounding the limit cycle.

In Fig. 11, we display a histogram of synchronization timesfor a chain of 500 relaxation oscillators. The histogram indicatesthat the distribution of synchronization times has the rough ap-pearance of a log-normal distribution if one ignores the peri-odic structure and examines its envelope. The periodic structurearises because we determine that a system is synchronous whenall oscillators jump at the same time. Given random initial con-ditions, the first oscillator to jump occurs shortly after the starttime and the synchronization time frequently occursnear an integer interval of the period after that.

The data presented give the impression that is directly re-lated to . While it may be true in general that increases as

increases, it is not true that is the only factor controllingthe rate of synchrony. Different parameters can yield identicalvalues for , but the resultant scaling relation betweenand can appear quite different. Using , , and

, results in an average time to synchrony that increaseslogarithmically with the system size, even though ,a value nearly identical to one of the parameter sets shown in

Fig. 10 that exhibits a power law scaling relation. We do nothave a detailed understanding of which parameters are relatedto .

We examined the scaling relation for a variety of couplingstrengths and parameters. All parameters tested withexhibited a logarithmic scaling relation, while those with largervalues of exhibited either logarithmic or power law scaling,with a tendency toward power law scaling.

For system sizes from – , our numerical results are con-sistent with , though we do not know if this relationaccurately reflects the asymptotic behavior of the system. Thispower law scaling is very curious. For systems that exhibit thisscaling relation we observe “defects,” or spatial arrangements ofoscillators resembling traveling waves, arise that hinder the for-mation of synchrony. These defects have the same frequency astraveling waves and they can remain stable for a short amountof time (several periods) until they merge with a neighboringsynchronous cluster. We suspect that they may be the cause ofthe power law scaling. Because traveling waves occur with de-creasing frequency as decreases, and because they cannotexist unless the right branch represents a finite portion of theperiod, it is likely that the logarithmic scaling relation occursfor , and power law scaling for . Unfortunately,we have no theoretical explanations for the power law scalingat this time and refer the reader to [5], which describes travelingwaves, specific spin-wave type initial conditions which result insomewhat longer times to synchrony, and other relevant issues.

VIII. EFFECTS OF COUPLING FORM ON

RELAXATION OSCILLATORS

We now examine the synchronization rate in chains of relax-ation oscillators for and with two different couplings, aHeaviside function and a smooth sigmoid.

A. Heaviside Interaction

In Sections II–VII, relaxation oscillators were in the singularlimit and no time was needed for an oscillator to jump frombranch to branch. We now investigate the case of whichresults in a finite time needed for an oscillator to jump frombranch to branch. This immediately causes a fundamentalchange in that the time needed for one oscillator to induceanother oscillator to jump is finite and information can onlypropagate from one end of the chain to the other at timesproportional to [39].

In Fig. 12, we present our data for relaxation oscillators in thesinusoidal regime with a Heaviside coupling in theform of histograms of the time to synchrony divided by . Datafrom two chain lengths and , and two values of

( and ) are shown. Although the histograms arenoisy and the variance of the time to synchrony is quite large,the scaled histograms have a qualitative match in their shape andtheir extent. The values of and its standard deviation bothappear to increase linearly with the system size for bothand . A value of places the relaxation oscillatorsclearly in the sinusoidal regime and a value of arguablyplaces the relaxation oscillators in the sinusoidal regime.


Fig. 12. Histograms of the time to synchrony divided by n using a nearlydiscontinuous interaction (� = 5000) for chains of length n = 25 (thin)and n = 50 (thick) (a) Scaled histograms for " = 1:0 are based on 2000trials for both chain lengths. (b) Scaled histograms for " = 0:1 are based on1200 and 1500 trials for chain lengths of 25 and 50, respectively. The remainingparameters are � = 6, � = 3, = 42, � = �0:5, and � = 1000. Each trialwas performed with the initial conditions uniformly and randomly distributedon the lower left branch of the limit cycle.

For , a network is synchronous when the average Eu-clidean distance squared between the oscillators is lessthan 0.01

(18)

The limit cycle varies in both the - and -directions by roughly, thus, a threshold of 0.01 for indicates that the oscil-

lators are relatively close to each other. We tested several mea-sures of synchrony and did not see any discernible difference inour results. An adaptive fifth-order Runge–Kutta method from[36] generated all results for (see [5] for more details).

For completeness, we examined pairs of relaxation oscillatorsfor a few parameters with and found that there is a transi-tion from synchrony to aperiodicity between and .

B. Smooth Interaction

In this section, we set and the interaction is a smoothsigmoid instead of an approximate Heaviside step function.Fig. 13(a) displays histograms of the time to synchrony dividedby , for , and for chains of length and .It is evident that this scaling is appropriate. Fig. 13(b) and(c) displays similar histograms with andand for these values of , the scaled histograms do not line upprecisely. Scaling by overestimates the time to synchronyby a small but noticeable amount. This is as expected becausethe interaction is a nontrivial function of the coordinate andbecomes less smooth as decreases.

Our data confirms the numerical results of Somers and Kopellin that relaxation oscillator chains synchronize faster as de-creases [39], and in that greater slopes of the interaction term aidsynchronization [40]. In Table III, we show a subset of our datafor two extreme values of and several values of . Intermediatevalues of do not yield readily quantified scaling behaviors, butthey are intermediate between and . The data are also con-sistent with a time to synchrony proportional to . This is inagreement with perturbation analysis for relaxation oscillators,in which an expansion in gives a first term of order [2].

Our data may include large errors due to boundary effects thatare significant in comparison to the small system sizes (

Fig. 13. Histograms of the time to synchrony divided by n using a smoothinteraction (� = 1) for chains of length n = 25 (thin) and n = 50 (thick). (a)The scaled histograms for " = 1:0 are based on 1000 and 800 trials for chainlengths of 25 and 50, respectively. (b) Scaled histograms for " = 0:1 are basedon 1000 and 400 trials for chain lengths of 25 and 50, respectively. (c) Scaledhistograms for " = 0:01 are based on 4000 and 2500 trials for chain lengths of25 and 50, respectively. The remaining parameters are � = 6, � = 3, = 42,� = �0:5, and � = 1000. Each trial was performed with initial conditionsrandomly and uniformly distributed on the lower left branch of the limit cycle.

TABLE IIITHE AVERAGE TIME TO SYNCHRONY (IN PERIODS) FOR DIFFERENT VALUES OF

n, ", AND �. (A) CONTAINS OUR DATA FOR � = 1:0 AND (B) CONTAINS OUR

DATA FOR � = 5000. THE DATA SUGGEST AN INCREASE IN THE TIME TO

SYNCHRONY AS n FOR � = 1:0 AND AS n FOR � = 5000. THE DATA ALSO

SUGGEST THAT THE TIME TO SYNCHRONY IS PROPORTIONAL TO "

is the largest), or the effects of correlations that are larger thanthe size of the system used. In spite of these possibilities, wehave good reasons to believe in the scaling relations suggestedby our data. One reason is that the histograms for differenthave very similar shapes, indicating that these shapes result fromthe system parameters and not the size of the network. Also, ourdata indicate an expected proportionality to , and this pro-portionality might not be evident if boundary effects are signif-icant. Finally our data are consistent with the results reported in[39], [40], and [46].

It has been shown, that under certain conditions, many formsof oscillators and interactions can be reduced to a system ofphase oscillators coupled through a function of their phase dif-ferences [11]. To a first-order approximation, this becomes asystem of phase oscillators with diffusive coupling, and the timeto synchrony is then proportional to the length of the chainsquared (more general and complete arguments for this scalingrelation are made in [11], [22], [39]). This is a possible expla-nation for the observed scaling relationship, , whenthe interaction term is a smooth sigmoid.


Fig. 14. Plot of hT i=� as a function as a function of log (2L � 1) forL � L networks of locally coupled relaxation oscillators. All oscillators arecoupled with their four nearest neighbors, except those on the borders and thecorners, which are connected with three and two neighbors, respectively. Thecoupling is normalized as described in Section IV. (a) Network parameters aresuch that B = 0:021 (� = 8, � = 112, and = 115), and the averages arebased on 300 trials except for the average of the largest square which is basedon 100 trials. (b) Network parameters are such thatB = 0:33 (� = 8, � = 8,and = 11), and the averages are based on 250 or more trials except for thelargest two square sizes, which are based on 150 and 14 trials, respectively. Theerror bars are shown and indicate the standard deviation. The initial conditionswere randomly and uniformly chosen to lie on the lower left branch of the limitcycle to avoid nonsynchronous solutions.

Based on this data, we conjecture that a discontinuous interac-tion has better properties of synchronization than a smooth inter-action for many types of oscillators. This conjecture is in agree-ment with the work of Daido [10] in which a step-like interactionperfectly synchronizes a portion of the oscillators in a globallycoupled network with a normal distribution of intrinsic frequen-cies, whereas a smooth coupling cannot. Also, chains of pulsecoupled spike oscillators synchronize at times proportional to

[7], but when weakly coupled with a smoothly varyingfunction, the scaling relation changes completely [22]. Thesedata indicate that a discontinuous interaction may be genericallybetter at producing quick synchronization than a smooth inter-action.

IX. SYNCHRONIZATION IN 2-D NETWORKS OF RELAXATION

OSCILLATORS IN THE SINGULAR LIMIT

Because synchrony was quickly attained in chains of oscilla-tors, we examined 2-D locally coupled networks also. Here weonly examined relaxation oscillators in the singular limit and inthe spiking regime because they can be numerically integratedrapidly [25] enabling us to present data for large network sizes.

In Fig. 14(a), we display as a function offor square networks of length varying from 4

to 250, with . Our data are consistent with. This scaling relation is the same as that observed in 2-D

networks of spike oscillators [7], thus, again indicating that re-laxation oscillators in the spiking regime appear to be funda-mentally similar to spike oscillators.

In rings of oscillators, traveling waves can occur. In 2-Dnetworks with periodic boundary conditions, traveling wavesand other more complex patterns occur. Rotating waves wereobserved but only when initial conditions were randomlydistributed within the box bounding the limit cycle (see [5] formore details). When we restricted the initial conditions to thelower left branch of the limit cycle, synchrony was always thefinal solution.

As with traveling waves, rotating waves in 2-D lattices need ahighly specific positioning on the limit cycle to occur and are notobserved frequently when initial conditions are randomly gen-erated. This is explained by the fact that a rotating wave needsat least two or more oscillators or groups of oscillators to beactive at any given time. As the time spent on the right branchdecreases, the temporal ordering of the oscillators must becomemore and more precise, and the number of oscillators requiredfor a rotating wave increases. The frequency with which rotatingwaves are observed decreases as decreases. We refer thereader to [5] for a more complete description of this behavior.Note that traveling or rotating waves were not observed in 1-Dand 2-D locally coupled networks of spike oscillators [7].

In Fig. 14(b), we display as a function offor square networks with varying from 2 to 500

with . Even though synchrony is always achieved,there is no obvious scaling behavior. The curve is nonmonotonicand nonintuitive. We have no theoretical understanding of howthis curve arises. Note the exceptionally short synchronizationtimes.

X. DISCUSSION

Synchronization phenomena are widely observed in nature,and are the subject of intense research. Thus far, the study ofcoupled oscillator systems has revealed a variety of behaviors,including synchrony [18], [31], phase transitions [24], [41], par-tial synchrony [10], antiphase solutions [23], traveling waves[14], [40], cluster formation and dissipation [12], and self-orga-nized criticality [4]. The complexity of oscillator systems arisesin part from the variety of their components, and determiningfactors include the nature of the oscillator, the form of the inter-action (as well as its amplitude, time delay, spatial extent, andhomogeneity), the distribution of intrinsic frequencies, externaldriving forces, and noise.

We numerically examined how the rate of synchrony changesin networks of relaxation oscillators as we changed the natureof the oscillation. Four different classes of relaxation oscillatorwere studied with a Heaviside coupling, and three different syn-chronization rates were observed . was observedfor relaxation oscillators in the spiking regime for chains oflength and for 2-D square networks of length

. was observed for relaxation oscillators in thesingular limit. was observed for relaxation oscillatorsand relaxation oscillators in the sinusoidal regime. These results


indicate that the kind of the oscillation changes the rate of syn-chronization.

We note that for small values of , the function appearsas a straight line on a semilog plot. Therefore, canbe hard to distinguish from logarithmic scalingfor even relatively large chain lengths. From our data we cannotexactly determine whether or not the scaling relation is loga-rithmic or a power law with small .

For relaxation oscillator chains with , when the cou-pling changed from a nearly discontinuous step function to asmooth sigmoid, the synchronization rate changed from

to . These results indicate that the form of the cou-pling modifies the synchronization rate.

All of our numerical results were performed using finite sys-tems. Although the aforementioned trends are apparent in ourdata, we do not know with certainty that they accurately reflectthe asymptotic behavior of the systems studied.

Somers and Kopell [39] emphasized the role of relaxation os-cillations in obtaining quick synchrony and suggested that “fastthreshold modulation” (FTM) [39], [40] is responsible for thelinear scaling relationship. Though they used a Morris–Lecaroscillator with a sigmoid interaction term multiplied by thevoltage-like variable, the same basic dynamics of FTM areobserved with Terman–Wang oscillators in the relaxationregime. For Terman–Wang oscillators in the sinusoidal regime,the dynamics of FTM are not observed and the linear scalingrelation persists, thus, suggesting that an additional mechanismexists that causes the linear scaling relationship.

We also note that the coupling term is a nontrivial function ofthe fast variable. When the position of the oscillator changesrapidly, the interaction term exhibits rates of change accord-ingly. In this sense, the nature of the relaxation oscillation causesthe interaction term to become step-like. Our data suggest thatas becomes smaller the system exhibits synchronization timesthat are shorter than , but we do not have a good es-timate for what value of is required for a relaxation oscillatornetwork of a given size and interaction to exhibit .

We conjecture that a discontinuous interaction has betterproperties of synchronization than smooth interactions in manyoscillator systems. This is consistent with Daido’s argumentthat a step-like coupling “embodies a limit of strong couplingbecause it yields finite synchronizing force even from an in-finitesimal phase difference” [10] (see also [21]). We note thatanother benefit of a step-like coupling is that synchronizationoccurs quicker.

Our conjecture may have practical applications. Resonancetunneling diodes [8], [50] oscillate at megahertz and higher fre-quencies. However, the output current of these devices is verysmall and there is a need to couple many diodes together tocreate devices that yield high-output currents at these frequen-cies. Similarly, phase synchrony of Josephson arrays is desirable[15]. At the moment, it is unknown how to quickly synchronizethe outputs of locally coupled arrays of these devices. Even apartial understanding of how to achieve quick synchrony in thepresence of noise and disorder would be valuable. Step-like, orpulsatile interactions may also be useful in synchronization ofchaotic systems (see [26], [42] for example).

While emphasizing the flexibility of relaxation oscillators, wehave only discussed a limited set of relaxation oscillator systemsand our results are focussed almost entirely on synchronization.Networks of relaxation oscillators can exhibit a variety of be-haviors. With inhibitory coupling, our experiments showed thatchains of oscillators quickly attained an entrained state. For re-laxation oscillators with an intermediate time scale on the activephase of the limit cycle, slower than the transition time scale,but faster than the slow time scale, almost synchronous solu-tions arise, and desynchronous solutions exist also [3]. Withtime-delay coupling, chains of Terman–Wang relaxation oscil-lators with Heaviside coupling exhibit a rapid approach to aloosely synchronous solution [6], although antiphase solutionsexist dependent on the coupling strength and the time delay.For other relaxation oscillators, the synchronous solution canbe stable even in the presence of time delays [13]. For a pairof piece-wise linear relaxation oscillators in the singular limit,in-phase and antiphase solutions arise dependent on the initialconditions, the rate of decay of the interaction, and the couplingstrength [37] (also see [9]). With diffusive coupling, lattices ofrelaxation oscillators can synchronize but the coupling strengthmust be large enough [1]. In a chain of relaxation oscillatorswith different intrinsic frequencies and diffusive coupling, syn-chrony is attained, but exhibits extremely long transients evenin the presence of large coupling [29]. These behaviors revealthe complexity and subtlety of the systems involved.

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Shannon R. Campbell received the B.S. degree inphysics and mathematics from the University of Cali-fornia, Davis, in 1990 and the Ph.D. degree in physicsfrom The Ohio State Univerity, Columbus, in 1997.

From 1997 to 2004, he held various positions inindustry and at the National Institutes of Health,Bethesda, MD, working in image processing andpattern recognition applications. He is currentlysupported by SBIR Grants and his research interestsare in pattern recognition, image processing, neuro-dynamics, and human visual perception.

DeLiang Wang (M’90–F’04) received the B.S. andM.S. degrees in computer science from Peking (Bei-jing) University, Beijing, China, in 1983 and 1986,respectively, and the Ph.D. degree from the Univer-sity of Southern California, Los Angeles, in 1991.

From 1986 to 1987, he was with the Institute ofComputing Technology, Academia Sinica, Beijing,China. Since 1991, he has been with the Departmentof Computer and Information Science and the Centerfor Cognitive Science, The Ohio State University,Columbus, where he is currently a Professor. From

1998 to 1999, he was a Visiting Scholar in the Department of Psychology,Harvard University, Cambridge, MA. His research interests include machineperception and neurodynamics.

Dr. Wang currently chairs the IEEE Neural Networks Society Neural Net-works Technical Committee, is a member of the Governing Board of the Inter-national Neural Network Society and the IEEE Signal Processing Society Ma-chine Learning for Signal Processing Technical Committee. He is a recipient ofthe U.S. Office of Naval Research Young Investigator Award.

Ciriyam Jayaprakash, photograph and biography not available at the time ofpublication.

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